Bull. Korean Math. Soc. 2016; 53(2): 495-506
Printed March 31, 2016
https://doi.org/10.4134/BKMS.2016.53.2.495
Copyright © The Korean Mathematical Society.
V\'\i tor H. Fernandes and Teresa M. Quinteiro
Universidade NOVA de Lisboa, Instituto Superior de Engenharia de Lisboa
In this note we consider the monoid $\PODI_n$ of all monotone partial permutations on $\{1,\ldots,n\}$ and its submonoids $\DP_n$, $\POI_n$ and $\ODP_n$ of all partial isometries, of all order-preserving partial permutations and of all order-preserving partial isometries, respectively. We prove that both the monoids $\POI_n$ and $\ODP_n$ are quotients of bilateral semidirect products of two of their remarkable submonoids, namely of extensive and of co-extensive transformations. Moreover, we show that $\PODI_n$ is a quotient of a semidirect product of $\POI_n$ and the group $\mathcal{C}_2$ of order two and, analogously, $\DP_n$ is a quotient of a semidirect product of $\ODP_n$ and $\mathcal{C}_2$.
Keywords: transformations, partial isometries, order-preserving, semidirect products, pseudovarieties
MSC numbers: 20M20, 20M07, 20M10, 20M35
2017; 54(4): 1373-1386
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